Summary: SOLUTIONS OF STOCHASTIC DELAY EQUATIONS IN SPACES OF
CONTINUOUS FUNCTIONS
JAN VAN NEERVEN AND MARKUS RIEDLE
Dedicated to Rainer Nagel on the occasion of his 65th birthday
Abstract. We present a semigroup approach to stochastic delay equations of the
form
dX(t) =
0
-h
X(t + s) dµ(s) dt + dW(t) for t 0,
X(t) = f(t) for t [-h, 0],
in the space of continuous functions C[-h, 0]. We represent the solution as a C[-h, 0]-
valued process arising from a stochastic weak-integral in the bidual C[-h, 0] and
show how this process can be interpreted as a mild solution of an associated sto-
chastic abstract Cauchy problem. We obtain a necessary and sufficient condition
guaranteeing the existence of an invariant measure.
1. Introduction
In this paper we study the stochastic linear delay differential equation
dX(t) =
0